Circles

A circle is the locus of points in a plane at a constant distance from a fixed point

Angle Measure Theorem: ∠POQ = 2∠PRQ

Angle in a semicircle = 90°

If a cyclic arc subtends 90° at the circumference, then it is a semicircle

Angles in the same segment are equal

∠A + ∠C = 180° , ∠B + ∠D = 180°

Exterior angle of cyclic quadrilateral = opposite interior angle

If opposite angles of a quadrilateral sum to 180°, then it is cyclic

If exterior angle of a quadrilateral equals opposite interior angle, then the quadrilateral is cyclic

Radius through the point of contact is perpendicular to the tangent

A line drawn through the end of a radius and perpendicular to it is a tangent

Only one tangent can be drawn to a circle at a given point on the circle

If a point lies outside the circle, two tangents can be drawn from this point to the circle

If a point lies inside the circle, no tangent can be drawn from this point